3 Answers
3

Remember that a square matrix $B$ is the inverse of a square matrix $A$ if $AB = I$ (or $BA = I$; each one implies the other). Using the equation for $A$, can you show that $A(3I - A)$ or $(3I - A)A$ is equal to the identity matrix?

As you state, you've assumed $A^{-1}$ exists. How do you know a square matrix $A$ satisfying $A^2 - 3A + I = O$ has an inverse? What you've shown is that if $A$ has an inverse and $A$ satisfies $A^2 - 3A + I = 0$, then its inverse is $3I - A$.
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Michael AlbaneseSep 20 '12 at 5:08

Looks good to me (as long as you meant to put a comma between the $2$ and the $I$). Make sure that you can use the fact that (for square matrices) it is enough to show that $AB = I$, you may not have seen that result if this is homework; if you haven't seen the result, you can also rearrange the equation so that $A$ is on the right.
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Michael AlbaneseSep 20 '12 at 5:19

Meaning they commute? Thank you very much Michael!
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diimensionSep 20 '12 at 5:25